Metric entropy and the number of periodic orbits for endomorphisms

Abstract

Using the inverse limit technique, we demonstrate that for an ergodic hyperbolic measure μ preserved by a $C^2$ endomorphism f, the exponential growth rate of the number of periodic measures that approximate μ and that their corresponding Lyapunov exponents approximate the Lyapunov exponents of μ, equals the metric entropy $h_{\mu }\left(f\right)$.